<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="int_2.html#V1">prime</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><a NAME="E2:1_1"/><b>then </b><i><font color="Green">E3</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="idea_1.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E3:1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E5:1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="idea_1.html#E4:1_1_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<a NAME="E6:1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E7:1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E8:1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="idea_1.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="idea_1.html#E4:1_1_1"><i><font color="Green">E4</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="idea_1.html#E2:1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>


</div>
